Let us recall the \({\rm U}(1)\) -current net \(({{\mathcal {A}}^{(0)}}, U_0, \Omega _0)\) , which is a conformal net. For its definition,
see our previous discussion [1]}.
The point is that the Hilbert space is naturally isomorphic to the Fock space \(\mathcal {H}_{\mathrm {r}}\) 
of the massive free net, whose one-particle space is \(L^2({\mathbb {R}},d\theta )\)  and
one considers the second quantization operators.
The above one-dimensional triple \((\mathcal {M}_{\mathrm {r}}, U_{\mathrm {r}}^+, \Omega _{\mathrm {r}})\)  is actually unitarily equivalent
to the triple \(({{\mathcal {A}}^{(0)}}({\mathbb {R}}_+), T_0, \Omega _0)\) ,
where \(({{\mathcal {A}}^{(0)}}, U_0, \Omega _0)\)  is the \({\rm U}(1)\) -current net and \(T_0\)  is the restriction
of \(U_0\)  to the translation subgroup. This will be explained
in more detail in [2]}.
